Analiza minimalno potrebnog koeficijenta trenja klizanja pri brahistohronom kretanju neholonomnog mehaničkog sistema
Analysis of the minimum required coefficient of sliding friction at brachistochronic motion of a nonholonomic mechanical system
Abstract
Analizira se problem brahistohronog kretanja mehaničkog sistema na primeru jednog uprošćenog modela vozila. Sistem se kreće između dva zadata položaja pri neizmenjenoj vrednosti mehaničke energije u toku kretanja. Diferencijalne jednačine kretanja, u kojima figurišu reakcije neholonomnih veza i upravljačke sile, dobijaju se na osnovu opštih teorema dinamike. Ovde je to podesnije umesto nekih drugih metoda analitičke mehanike primenjenih na neholonomne sisteme, u kojima je neophodno dati naknadno fizičko tumačenje množitelja veza da bi se ovaj problem rešio. Podesnim izborom veličina stanja, dobija se, najprostiji moguć u ovom slučaju, zadatak optimalnog upravljanja, koji se rešava primenom Pontrjaginovog principa maksimuma. Numeričko rešavanje dvotačkastog graničnog problema vrši se metodom šutinga. Na osnovu tako dobijenog brahistohronog kretanja određuju se aktivne upravljačke sile, a ujedno i reakcije veza. Koristeći Kulonove zakone trenja klizanja, određuje se minimalno potrebna vr...ednost koeficijenta trenja klizanja, da ne bi došlo do proklizavanja vozila u tačkama kontakta sa podlogom.
The paper analyzes the problem of brachistochronic motion of a nonholonomic mechanical system, using an example of a simple car model. The system moves between two default positions at an unaltered value of the mechanical energy during motion. Differential equations of motion, containing the reaction of nonholonomic constraints and control forces, are obtained on the basis of general theorems of dynamics. Here, this is more appropriate than some other methods of analytical mechanics applied to nonholonomic systems, where the provision of a subsequent physical interpretation of the multipliers of constraints is required to solve this problem. By the appropriate choice of the parameters of state as simple a task of optimal control as possible is obtained in this case, which is solved by the application of the Pontryagin maximum principle. Numerical solution of the two-point boundary value problem is obtained by the method of shooting. Based on the thus acquired brachistochronic motion, t...he active control forces are determined as well as the reaction of constraints. Using the Coulomb laws of friction sliding, the minimum value of the coefficient of friction is determined to avoid car skidding at the points of contact with the ground.
Keywords:
Pontryagin's maximum principle / Optimal control / Nonholonomic mechanical system / Coulomb friction / BrachistochroneSource:
FME Transactions, 2014, 42, 3, 199-204Publisher:
- Univerzitet u Beogradu - Mašinski fakultet, Beograd
Funding / projects:
- Micromechanical criteria of damage and fracture (RS-MESTD-Basic Research (BR or ON)-174004)
- Sustainability and improvement of mechanical systems in energetic, material handling and conveying by using forensic engineering, environmental and robust design (RS-MESTD-Technological Development (TD or TR)-35006)
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Institution/Community
Mašinski fakultetTY - JOUR AU - Radulović, Radoslav AU - Obradović, Aleksandar AU - Jeremić, Bojan PY - 2014 UR - https://machinery.mas.bg.ac.rs/handle/123456789/1972 AB - Analizira se problem brahistohronog kretanja mehaničkog sistema na primeru jednog uprošćenog modela vozila. Sistem se kreće između dva zadata položaja pri neizmenjenoj vrednosti mehaničke energije u toku kretanja. Diferencijalne jednačine kretanja, u kojima figurišu reakcije neholonomnih veza i upravljačke sile, dobijaju se na osnovu opštih teorema dinamike. Ovde je to podesnije umesto nekih drugih metoda analitičke mehanike primenjenih na neholonomne sisteme, u kojima je neophodno dati naknadno fizičko tumačenje množitelja veza da bi se ovaj problem rešio. Podesnim izborom veličina stanja, dobija se, najprostiji moguć u ovom slučaju, zadatak optimalnog upravljanja, koji se rešava primenom Pontrjaginovog principa maksimuma. Numeričko rešavanje dvotačkastog graničnog problema vrši se metodom šutinga. Na osnovu tako dobijenog brahistohronog kretanja određuju se aktivne upravljačke sile, a ujedno i reakcije veza. Koristeći Kulonove zakone trenja klizanja, određuje se minimalno potrebna vrednost koeficijenta trenja klizanja, da ne bi došlo do proklizavanja vozila u tačkama kontakta sa podlogom. AB - The paper analyzes the problem of brachistochronic motion of a nonholonomic mechanical system, using an example of a simple car model. The system moves between two default positions at an unaltered value of the mechanical energy during motion. Differential equations of motion, containing the reaction of nonholonomic constraints and control forces, are obtained on the basis of general theorems of dynamics. Here, this is more appropriate than some other methods of analytical mechanics applied to nonholonomic systems, where the provision of a subsequent physical interpretation of the multipliers of constraints is required to solve this problem. By the appropriate choice of the parameters of state as simple a task of optimal control as possible is obtained in this case, which is solved by the application of the Pontryagin maximum principle. Numerical solution of the two-point boundary value problem is obtained by the method of shooting. Based on the thus acquired brachistochronic motion, the active control forces are determined as well as the reaction of constraints. Using the Coulomb laws of friction sliding, the minimum value of the coefficient of friction is determined to avoid car skidding at the points of contact with the ground. PB - Univerzitet u Beogradu - Mašinski fakultet, Beograd T2 - FME Transactions T1 - Analiza minimalno potrebnog koeficijenta trenja klizanja pri brahistohronom kretanju neholonomnog mehaničkog sistema T1 - Analysis of the minimum required coefficient of sliding friction at brachistochronic motion of a nonholonomic mechanical system EP - 204 IS - 3 SP - 199 VL - 42 DO - 10.5937/fmet1403199R ER -
@article{ author = "Radulović, Radoslav and Obradović, Aleksandar and Jeremić, Bojan", year = "2014", abstract = "Analizira se problem brahistohronog kretanja mehaničkog sistema na primeru jednog uprošćenog modela vozila. Sistem se kreće između dva zadata položaja pri neizmenjenoj vrednosti mehaničke energije u toku kretanja. Diferencijalne jednačine kretanja, u kojima figurišu reakcije neholonomnih veza i upravljačke sile, dobijaju se na osnovu opštih teorema dinamike. Ovde je to podesnije umesto nekih drugih metoda analitičke mehanike primenjenih na neholonomne sisteme, u kojima je neophodno dati naknadno fizičko tumačenje množitelja veza da bi se ovaj problem rešio. Podesnim izborom veličina stanja, dobija se, najprostiji moguć u ovom slučaju, zadatak optimalnog upravljanja, koji se rešava primenom Pontrjaginovog principa maksimuma. Numeričko rešavanje dvotačkastog graničnog problema vrši se metodom šutinga. Na osnovu tako dobijenog brahistohronog kretanja određuju se aktivne upravljačke sile, a ujedno i reakcije veza. Koristeći Kulonove zakone trenja klizanja, određuje se minimalno potrebna vrednost koeficijenta trenja klizanja, da ne bi došlo do proklizavanja vozila u tačkama kontakta sa podlogom., The paper analyzes the problem of brachistochronic motion of a nonholonomic mechanical system, using an example of a simple car model. The system moves between two default positions at an unaltered value of the mechanical energy during motion. Differential equations of motion, containing the reaction of nonholonomic constraints and control forces, are obtained on the basis of general theorems of dynamics. Here, this is more appropriate than some other methods of analytical mechanics applied to nonholonomic systems, where the provision of a subsequent physical interpretation of the multipliers of constraints is required to solve this problem. By the appropriate choice of the parameters of state as simple a task of optimal control as possible is obtained in this case, which is solved by the application of the Pontryagin maximum principle. Numerical solution of the two-point boundary value problem is obtained by the method of shooting. Based on the thus acquired brachistochronic motion, the active control forces are determined as well as the reaction of constraints. Using the Coulomb laws of friction sliding, the minimum value of the coefficient of friction is determined to avoid car skidding at the points of contact with the ground.", publisher = "Univerzitet u Beogradu - Mašinski fakultet, Beograd", journal = "FME Transactions", title = "Analiza minimalno potrebnog koeficijenta trenja klizanja pri brahistohronom kretanju neholonomnog mehaničkog sistema, Analysis of the minimum required coefficient of sliding friction at brachistochronic motion of a nonholonomic mechanical system", pages = "204-199", number = "3", volume = "42", doi = "10.5937/fmet1403199R" }
Radulović, R., Obradović, A.,& Jeremić, B.. (2014). Analiza minimalno potrebnog koeficijenta trenja klizanja pri brahistohronom kretanju neholonomnog mehaničkog sistema. in FME Transactions Univerzitet u Beogradu - Mašinski fakultet, Beograd., 42(3), 199-204. https://doi.org/10.5937/fmet1403199R
Radulović R, Obradović A, Jeremić B. Analiza minimalno potrebnog koeficijenta trenja klizanja pri brahistohronom kretanju neholonomnog mehaničkog sistema. in FME Transactions. 2014;42(3):199-204. doi:10.5937/fmet1403199R .
Radulović, Radoslav, Obradović, Aleksandar, Jeremić, Bojan, "Analiza minimalno potrebnog koeficijenta trenja klizanja pri brahistohronom kretanju neholonomnog mehaničkog sistema" in FME Transactions, 42, no. 3 (2014):199-204, https://doi.org/10.5937/fmet1403199R . .